This is a continuation of a series of papers on geometric mapping theory [1]}, [2]}, [3]}, [4]}, [5]}, in which we establish (partial) rigidity and (partial) regularity of bilipschitz, quasiconformal, or more generally Sobolev mappings in the subriemannian setting (in particular in Carnot groups). In [1]} we showed that for Sobolev maps between Carnot groups, the pullback of differential forms using the Pansu differential partially respects exterior differentiation; this result served as the starting point for a variety of applications.
Our aim in this paper is to relate the Pullback Theorem with the spectral sequence considered by Rumin in [7]}; our main result suggests that the spectral sequence provides a natural framework for interpreting the Pullback Theorem.
